Critical points, maximization and minimization problems

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Let $f(x) =e^{- \frac{x^{3}}{3} + 2 x^{2} - 4 x + 2} $.
1. Calculate $f'(x)$.
2. Find the critical points of $f$.
3. Find the local minima and maxima (or local extrema) of $f$. For each extremum, determine three things: its location (i.e., value of $x$), its value (i.e., value of $f(x)$), and whether it is a local maximum or a local minimum.
4. Find the global maximum and global minimum of the function $f(x)$ on the interval $0 \le x \le 3$. Also indicate the location (the value of $x$) of the global maximum and global minimum.
5. Sketch the graph of $f(x)$ on the interval $0 \le x \le 3$. Your sketch should be consistent with the information you determined.
Let $f(x) =x^{4} e^{- 2 x} $.
1. Calculate $f'(x)$.
2. Find the critical points of $f$.
3. Find the local minima and maxima (or local extrema) of $f$. For each extremum, determine three things: its location (i.e., value of $x$), its value (i.e., value of $f(x)$), and whether it is a local maximum or a local minimum.
4. Find the global maximum and global minimum of the function $f(x)$ on the interval $\frac{1}{2} \le x \le 3$. Also indicate the location (the value of $x$) of the global maximum and global minimum.
5. Sketch the graph of $f(x)$ on the interval $\frac{1}{2} \le x \le 3$. Your sketch should be consistent with the information you determined.
Let $f(x) =\left(- 2 x^{2} + 8 x + 10\right) e^{\frac{x}{4}} $.
1. Calculate $f'(x)$.
2. Find the critical points of $f$.
3. Find the local minima and maxima (or local extrema) of $f$. For each extremum, determine three things: its location (i.e., value of $x$), its value (i.e., value of $f(x)$), and whether it is a local maximum or a local minimum.
4. Find the global maximum and global minimum of the function $f(x)$ on the interval $-8 \le x \le 5$. Also indicate the location (the value of $x$) of the global maximum and global minimum.
5. Sketch the graph of $f(x)$ on the interval $-8 \le x \le 5$. Your sketch should be consistent with the information you determined.
Let $f(x) =- \left(x + 2\right) \left(x + 5\right) \left(x + 10\right) = - x^{3} - 17 x^{2} - 80 x - 100$.
1. Calculate $f'(x)$.
2. Find the critical points of $f$.
3. Determine on which intervals $f$ is increasing and $f$ is decreasing.
4. What are the roots of $f$ itself, i.e., at what points is $f(x)=0$?
5. Sketch the graph of $f(x)$. Your sketch should be consistent with the information you determined.

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Critical points, maximization and minimization problems

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Math 1241, Fall 2020